The control of the controller: molecular mechanisms for robust perfect adaptation and temperature compensation.

Centre for Organelle Research, University of Stavanger, Stavanger, Norway.

Abstract

Organisms have the property to adapt to a changing environment and keep certain components within a cell regulated at the same level (homeostasis). "Perfect adaptation" describes an organism's response to an external stepwise perturbation by regulating some of its variables/components precisely to their original preperturbation values. Numerous examples of perfect adaptation/homeostasis have been found, as for example, in bacterial chemotaxis, photoreceptor responses, MAP kinase activities, or in metal-ion homeostasis. Two concepts have evolved to explain how perfect adaptation may be understood: In one approach (robust perfect adaptation), the adaptation is a network property, which is mostly, but not entirely, independent of rate constant values; in the other approach (nonrobust perfect adaptation), a fine-tuning of rate constant values is needed. Here we identify two classes of robust molecular homeostatic mechanisms, which compensate for environmental variations in a controlled variable's inflow or outflow fluxes, and allow for the presence of robust temperature compensation. These two classes of homeostatic mechanisms arise due to the fact that concentrations must have positive values. We show that the concept of integral control (or integral feedback), which leads to robust homeostasis, is associated with a control species that has to work under zero-order flux conditions and does not necessarily require the presence of a physico-chemical feedback structure. There are interesting links between the two identified classes of homeostatic mechanisms and molecular mechanisms found in mammalian iron and calcium homeostasis, indicating that homeostatic mechanisms may underlie similar molecular control structures.

Scheme of integral control/feedback of a perturbed system, where the system output is perfectly adapted to the setpoint (i.e., the error e is robustly controlled to zero). MV and CV are the manipulated and controlled variables, respectively. Symbols in gray denote the notation for integral feedback by Yi et al. ().

(a) Reaction scheme of system with rate Eqs. . Species A is formed by a zero-order process with rate constant ksynth and then transformed to the product A1. Rate constant kpert is related to a perturbing process (wavy line), which increases the level of A. To remove excess A, A is forming enzyme species Eadapt, which removes A with the flux (indicated by the vertical arrow). To get robust adaptation in A independent of kpert, Eadapt is removed through a zero-order flux j0 = kadaptAset. (b) Calculation showing that negative Eadapt concentrations may arise when Aset is regarded as a fixed setpoint. Initial concentrations of A and Eadapt are zero; kadapt = 5, kpert = ksynth = 0.5, , = 1, = 110, = 100, and Aset = 2. Concentration and timescales are in arbitrary units (a.u.). (c) Scheme of the adaptive process shown in panel a containing the setpoint Aset, the integral controller and the process units. The controlled variable (CV) is A. Eqs. are written as dA/dt = f2(·) – f1(·) + kpert, dEadapt/dt = kadapt(A – Aset), respectively, with , , and = K · Eadapt. K is the turnover number for Eadapt, i.e., . MV: manipulated variable.

To avoid negative concentrations in Eadapt, j0 (a) is represented as an enzymatic zero-order process. (a) Fully expanded Michaelis-Menten mechanism. The rate equations together with rate constants are shown in the . To obtain robust homeostasis in A, Eset removes active Eadapt into an inactive form Eadapt∗ under zero-order conditions with Aset given by Eq. . (b) Same mechanism as in panel a, but formulating the Michaelis-Menten mechanism under steady-state/rapid equilibrium conditions. Rate equations are given in the . A zip-archive containing MATLAB and Berkeley Madonna versions of the model shown in Fig. 3 A with instructions and annotation available in the .

Robust perfect adaptation in A with Aset = 1.0. (a) Model described in a with rate constants as given in the . At t = 5.0 time units, kpert is increased from 0.0 to 1.0. (b) Initial conditions as given in the with kpert = 1.0. At t = 5.0 time units, kpert is increased from 1.0 to 103 a.u. (c) Initial conditions as in panel b. At t = 5.0 time units, kpert is decreased from 1.0 to 10−3 a.u. (d) Same initial conditions as in panel b, but Etr is successively increased leading eventually to the breakdown in homeostasis indicated by the decreasing Ass values. This breakdown can be opposed to a certain degree by increasing the values of kpert or ksynth. In the figure, kpert or ksynth were increased from their original values 1.0 and 3.0 to 10.0 and 12.0, respectively, thereby extending the homeostasis to larger Etr values (dashed line). However, at higher Etr concentrations the homeostasis fails again with decreasing Ass values (data not shown). (e) Calculated Ass values for varying with ksynth = 3.0 a.u. and kpert = 1.0 a.u. (solid circles), or with ksynth = 3.0 a.u. and kpert = 5.0 a.u. (open circles). For < 109 a.u., perfect homeostasis in A is lost (indicated by the condition that Ass < Aset), because for decreasing the associated with the removal of Eadapt by Eset increases, which eventually leads to the loss of the zero-order kinetics in the Eadapt degradation. (f) Time profiles in A with two different values. At t = 5.0 time units, kpert is increased from 1.0 to 5.0 a.u. 1 = Perfect homeostasis in A for = 1012 a.u.; 2 = Loss of perfect homeostasis in A when = 106 a.u., which is due to the loss of zero-order kinetics in the degradation of Eadapt.

Robust perfect temperature compensation of the model described in a. Rate constants () refer to 25°C with Aset = 1. (a) All activation energies are 50 kJ/mol and temperature is 5°C. (b) All activation energies are 50 kJ/mol and temperature is 100°C. Please note the much shorter timescale compared to panel a needed for the system to approach the same steady state. (c) Activation energies are as given in the . The system is initially at its steady state at 5°C. At 0.02 time units, the temperature is changed to 100°C showing perfect homeostasis only in A. (d) Activation energies as in panel c. The system is initially at its steady state at 100°C. At 40.0 time units the temperature is changed to 5°C showing perfect homeostasis only in A.

Homeostatic control motifs. Due to the kinetic restriction that concentrations need to be positive, two classes of homeostatic controllers arise: 1), inflow homeostatic controllers leading to homeostasis in the concentration of A for increasing (and moderate decreasing) perturbations in A (panels a and b); and 2), outflow homeostatic controllers leading to homeostasis in A for decreasing (and moderate increasing) perturbations in A (panels c and d). Rate equations with example parameter values are given in the . Note that many of the parameter values may be changed within certain limits (besides changing kpert) without affecting the homeostasis. (a) Schematic representation of the two (inflow) homeostatic models shown in . Robust homeostasis is due to the zero-order kinetic removal of Eadapt. (b) Inflow homeostatic model where Eadapt inhibits the inflow of A through ksynth. To maintain homeostasis the perturbation needs to be applied to the same reaction channel as ksynth. The integral feedback is due to the zero-order removal of Eadapt and is not related to the physico-chemical negative feedback from Eadapt to (ksynth + kpert). (c) Outflow homeostatic controller by removing Eadapt through A. (d) Outflow homeostatic controller by inhibiting Etr through Eadapt. Similar to panel b, homeostasis is obtained when the perturbation increases the outflow of A through the same reaction channel that is used by enzyme Etr.

Schematic representation of blood calcium homeostasis in humans. Important regulators are parathyroid hormone (PTH), calcitonin (CT), vitamin D, and the calcium-sensing receptor in the nephron. For a discussion of how these regulators may participate in inflow- and outflow mechanisms, see main text.